Is cos(5°) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).

Nov 1st 2006, 07:59 PM

ThePerfectHacker

Quote:

Originally Posted by harold

I have to answer the following and support it with a full proof.

Is cos(5°) rational? (Given hint: show that if cos(x) is rational, then cos(3x) is rational).

And,
So if is rational
Then, is also rational.
---
If is rational.
Then, is rational.
Then, is rational.
But that is not true.
Contradiction.

Nov 1st 2006, 08:46 PM

harold

Hi THP,

I was thinking the exact same thing by showing cos(45°) but it still doesn't prove it since we need to show if cos x is rational, then cos 3x is rational....then show cos 5° is irrat. based on the previous two proofs...?

Nov 1st 2006, 08:53 PM

ThePerfectHacker

Quote:

Originally Posted by harold

Hi THP,

I was thinking the exact same thing by showing cos(45°) but it still doesn't prove it since we need to show if cos x is rational, then cos 3x is rational....then show cos 5° is irrat. based on the previous two proofs...?

Yes it does.

If assume
Is rational and you arrived at a contradiction.

If is a rational number then, is a rational number.
Because when you add or multiply rationals you get a rational.